In all these computations we have used that \(\eta \) , \(\psi ^i\)  and
\(\dot{\psi }^i\)  anticommute.
We hence conclude that the quantity \(\left\langle \psi ,\dot{x}\right\rangle \)  is conserved, i.e. Noether's theorem holds herein a way similar to the fact that the time translation symmetry
\(t\longmapsto t+\varepsilon \)  induces the fields transformation \((x,\psi )\longmapsto (x - \varepsilon \dot{x},\psi - \varepsilon \nabla _{\dot{x}}\psi )\)  which is related, through Noether's theorem, to the conservation of the energy \({1\over 2}|\dot{x}|^2 + {1\over 2} \left\langle \psi , \nabla _{\dot{x}}\psi \right\rangle \) ., hence the symmetry which generates our fields transformation should have a geometrical interpretation. Eventually this `Noether's charge' has an interesting physical meaning, since the corresponding quantum operator is nothing but the Dirac operator on \(\mathcal {N}\)  (see [1]}, [2]}, [3]}, [4]}).
